The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
rev(++(x, y)) →⁺ ++(rev(y), rev(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [y / ++(x, y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Types:
rev :: a:b:++ → a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ → a:b:++ → a:b:++
hole_a:b:++1_0 :: a:b:++
gen_a:b:++2_0 :: Nat → a:b:++

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
rev

(8) Obligation:

TRS:
Rules:
rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Types:
rev :: a:b:++ → a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ → a:b:++ → a:b:++
hole_a:b:++1_0 :: a:b:++
gen_a:b:++2_0 :: Nat → a:b:++

Generator Equations:
gen_a:b:++2_0(0) ⇔ a
gen_a:b:++2_0(+(x, 1)) ⇔ ++(a, gen_a:b:++2_0(x))

The following defined symbols remain to be analysed:
rev

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
rev(gen_a:b:++2_0(+(1, 0)))

Induction Step:
rev(gen_a:b:++2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
++(rev(gen_a:b:++2_0(+(1, n4_0))), rev(a)) →_IH
++(*3_0, rev(a)) →_R^Ω(1)
++(*3_0, a)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Types:
rev :: a:b:++ → a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ → a:b:++ → a:b:++
hole_a:b:++1_0 :: a:b:++
gen_a:b:++2_0 :: Nat → a:b:++

Lemmas:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_a:b:++2_0(0) ⇔ a
gen_a:b:++2_0(+(x, 1)) ⇔ ++(a, gen_a:b:++2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Types:
rev :: a:b:++ → a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ → a:b:++ → a:b:++
hole_a:b:++1_0 :: a:b:++
gen_a:b:++2_0 :: Nat → a:b:++

Lemmas:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_a:b:++2_0(0) ⇔ a
gen_a:b:++2_0(+(x, 1)) ⇔ ++(a, gen_a:b:++2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)